A102413 → θ+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(-4\,x^{2} + n - n x + n x^{2} - n x^{3} - n^{2} + 3\,n^{2} x + n^{2} x^{2} + n^{2} x^{3} - 2\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(8\,x^{2} + n + 7\,n x + 3\,n x^{2} + n x^{3} - n^{2} - 5\,n^{2} x - 5\,n^{2} x^{2} - n^{2} x^{3} - 2\,n^{3} x - 6\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(3\,x^{2} + 19\,x^{3} + 9\,x^{4} + x^{5} + n x + 10\,n x^{2} + 26\,n x^{3} + 10\,n x^{4} + n x^{5} + 2\,n^{2} x^{2} + 12\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 1 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	1	8	3													12
3	1	12	18	4												35
4	1	16	48	32	5											102
5	1	20	90	120	50	6										287
6	1	24	144	304	240	72	7									792
7	1	28	210	616	770	420	98	8								2151
8	1	32	288	1088	1920	1632	672	128	9							5770
9	1	36	378	1752	4050	4860	3066	1008	162	10						15323
10	1	40	480	2640	7600	12024	10640	5280	1440	200	11					40356
11	1	44	594	3784	13090	26004	30338	20944	8514	1980	242	12				105547
12	1	48	720	5216	21120	50832	74704	67776	38016	13040	2640	288	13			274414
13	1	52	858	6968	32370	91884	164346	187824	137826	64740	19162	3432	338	14		709815
14	1	56	1008	9072	47600	156072	330848	461024	425376	260120	104720	27216	4368	392	15	1827888

Technical details

Node id: n-08e17bedc079010f · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-2c591a5df260a960

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1,  2;
  2: 1,  8,   3;
  3: 1, 12,  18,    4;
  4: 1, 16,  48,   32,    5;
  5: 1, 20,  90,  120,   50,    6;
  6: 1, 24, 144,  304,  240,   72,    7;
  7: 1, 28, 210,  616,  770,  420,   98,    8;
  8: 1, 32, 288, 1088, 1920, 1632,  672,  128,   9;
  9: 1, 36, 378, 1752, 4050, 4860, 3066, 1008, 162, 10;

%F Formula

The row polynomials satisfy -(n^2-n+(2*n^3-3*n^2+n)*x-(2*n^3+n^2+n-4)*x^2-(n^2-n)*x^3)*P(n,x) = -(n^2-n+(2*n^3+5*n^2-7*n)*x+(6*n^3+5*n^2-3*n-8)*x^2+(n^2-n)*x^3)*P(n-1,x) + (n*x+(2*n^2+10*n+3)*x^2+(12*n^2+26*n+19)*x^3+(2*n^2+10*n+9)*x^4+(n+1)*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n + (2*n^3 + 5*n^2 - 7*n)*x + (6*n^3 + 5*n^2 - 3*n - 8)*x^2 + (n^2 - n)*x^3)*P[n - 1, x] + (n*x + (2*n^2 + 10*n + 3)*x^2 + (12*n^2 + 26*n + 19)*x^3 + (2*n^2 + 10*n + 9)*x^4 + (n + 1)*x^5)*D[P[n - 1, x], x])/(-n^2 + n - (2*n^3 - 3*n^2 + n)*x + (2*n^3 + n^2 + n - 4)*x^2 + (n^2 - n)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - n + (2*n^3 + 5*n^2 - 7*n)*x + (6*n^3 + 5*n^2 - 3*n - 8)*x^2 + (n^2 - n)*x^3)*v[i - 1] + (n*x + (2*n^2 + 10*n + 3)*x^2 + (12*n^2 + 26*n + 19)*x^3 + (2*n^2 + 10*n + 9)*x^4 + (n + 1)*x^5)*deriv(v[i - 1], x))/(-n^2 + n - (2*n^3 - 3*n^2 + n)*x + (2*n^3 + n^2 + n - 4)*x^2 + (n^2 - n)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.